1. PDE Exercises
given is a solution of the homogeneous Cauchy problem, with , Let be sufficiently often continuously differentiable. Show without using any explicit formular for that the function Qg defined by where is the solution of the Cauchy problem for the inhomogeneous wave equation with right hand side g and zero initial conditions. We want to show that this implies \newline For n = 1 this gives an alternative solution of Example 1. This implies
By parameter dependent integrals;
By parameter dependent integrals; The integral above goes to zero by the homogeneous Cauchy problem solution. Therefore we can write
We only need the properties of . This should be seen as an exercise in parameter dependent integrals.
2. Examples of distributions and their order
Let us recall the criterion: T: linear T is a distribution compact , such that for with supp Where the last norm is the suprenum norm. (c=1, p = 0, indepedent of K.
Assume compact the integral above can be considered as C(K) and it is less than infinity (p=0, independent of K). M k-dim oriented subset and we can consider a k(top) form There is a trick:
In local coordinates open We use the pullback in fact, but we can sloppy write sum over open sets, where we have coordinates using partition of unity. We need a volume form or take the modulus of If we define, in local coordinates of the correct orientation then is well-defined independent of co-ordinates. We can then write the following Because we can do this trick. The integral above becomes C(K) and is finite, since has a finite number. This is indeed a distribution and clearly of order 0.